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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '6912.ia', 'ambient_counter': 209, 'ambient_order': 6912, 'ambient_tex': 'D_6.S_4^2', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 2, 'counter': 2069, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '6912.ia.144.pe1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '144.pe1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 144, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '48.29', 'subgroup_hash': 29, 'subgroup_order': 48, 'subgroup_tex': '\\GL(2,3)', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '6912.ia', 'aut_centralizer_order': None, 'aut_label': '144.pe1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '3456.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['36.jv1', '48.fe1'], 'contains': ['288.hp1', '432.jy1', '576.bq1'], 'core': '3456.a1', 'coset_action_label': None, 'count': 144, 'diagramx': None, 'generators': [40297548, 36933, 87100665, 93448860794, 87261754346], 'label': '6912.ia.144.pe1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.g1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '144.pe1', 'old_label': '144.pe1', 'projective_image': '3456.jo', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '144.pe1', 'subgroup_fusion': None, 'weyl_group': '24.12'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 3, 2, 2], 'aut_gens': [[29, 23], [55, 69], [55, 23], [61, 46], [12, 31], [24, 46]], 'aut_group': '48.48', 'aut_hash': 48, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 48, 'aut_permdeg': 6, 'aut_perms': [143, 127, 576, 126, 121], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 12, 1, 1], [3, 8, 1, 1], [4, 6, 1, 1], [6, 8, 1, 1], [8, 6, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times S_4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 12, 1], [3, 8, 1], [4, 6, 1], [6, 8, 1], [8, 6, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '24.12', 'commutator_count': 1, 'commutator_label': '24.3', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 29, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 12, 1, 1], [3, 8, 1, 1], [4, 6, 1, 1], [6, 8, 1, 1], [8, 6, 2, 1]], 'element_repr_type': 'Lie', 'elementary': 1, 'eulerian_function': 18, 'exponent': 24, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 2], [4, 1, 1]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '24.12', 'hash': 29, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 3], 'inner_gens': [[29, 17], [73, 23]], 'inner_hash': 12, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': True, 'inner_tex': 'S_4', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 2], [2, 3], [3, 2], [4, 1]], 'label': '48.29', 'linC_count': 2, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 2, 'linQ_dim': 4, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'GL(2,3)', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 7, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 8, 'number_divisions': 7, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 15, 'number_subgroup_classes': 16, 'number_subgroups': 55, 'old_label': None, 'order': 48, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 13], [3, 8], [4, 6], [6, 8], [8, 12]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [28], 'outer_gens': [[55, 23]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 8, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [3, 2], [4, 2]], 'representations': {'PC': {'code': 340659098868084616, 'gens': [1, 2, 3, 4], 'pres': [5, -2, -3, -2, 2, -2, 41, 272, 277, 72, 723, 188, 133, 58]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [27313555, 6823348]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [29, 23], 'family': 'GL'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [31, 55, 37]}, 'Perm': {'d': 8, 'gens': [5454, 583, 28776, 16577, 5167]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\GL(2,3)', 'transitive_degree': 8, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [6, 4, 6, 6, 12, 12, 12], 'aut_gens': [[7620480, 127370880, 43548276, 80205945, 87178291200, 47537280, 6274228129, 174356582400, 174400491626, 93532719813, 40279680], [39953733, 87091200, 83839371, 119783329, 87178328133, 47537280, 180627522425, 93405312000, 93492787129, 174443710533, 127370880], [83462400, 127370880, 120127371, 47183865, 180583640133, 79833600, 6307235148, 93405312000, 93525068087, 174483990213, 40279680], [123742080, 40279680, 7264451, 127025868, 6227057733, 47537280, 180671068025, 174356582400, 174403783669, 93445628613, 87091200], [3665733, 40279680, 83832131, 127017465, 87178291200, 79833600, 6314498324, 174356582400, 174436793367, 93445628613, 87091200], [3628800, 40279680, 120144463, 87483364, 87178291200, 3991680, 6354061729, 174356582400, 174436794856, 93445628613, 87091200], [123742080, 40279680, 7288783, 127031444, 87178291200, 47537280, 180671066745, 93405312000, 93452518246, 174396899013, 87091200], [83499333, 40279680, 4007407, 87471948, 6227020800, 43545600, 180710644129, 174356582400, 174400506856, 93445628613, 127370880]], 'aut_group': None, 'aut_hash': 3573172111550600594, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 27648, 'aut_permdeg': 30, 'aut_perms': [196574670567864430508783501430586, 40877630628039085319222822978450, 133434976328591841519997874192435, 146011444027425158868110239209139, 230397002173369809074358406742109, 133788664134100294958618438698383, 85880023044415839676471921604948], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 3, 2, 1], [2, 6, 2, 1], [2, 9, 2, 1], [2, 12, 1, 1], [2, 18, 2, 1], [2, 36, 1, 2], [2, 72, 1, 1], [2, 108, 1, 1], [2, 216, 1, 1], [3, 2, 1, 1], [3, 8, 1, 2], [3, 16, 1, 2], [3, 64, 1, 1], [3, 128, 1, 1], [4, 6, 1, 1], [4, 6, 2, 1], [4, 18, 1, 2], [4, 18, 2, 1], [4, 36, 1, 2], [4, 54, 1, 1], [4, 72, 1, 1], [4, 108, 1, 2], [4, 216, 1, 1], [6, 2, 1, 1], [6, 6, 1, 2], [6, 8, 1, 2], [6, 12, 2, 1], [6, 16, 1, 2], [6, 24, 1, 3], [6, 24, 2, 2], [6, 48, 1, 2], [6, 48, 2, 1], [6, 64, 1, 1], [6, 72, 1, 1], [6, 72, 2, 1], [6, 96, 1, 1], [6, 96, 2, 1], [6, 128, 1, 1], [6, 144, 1, 1], [6, 144, 2, 1], [6, 192, 1, 1], [6, 192, 2, 1], [6, 288, 1, 1], [8, 6, 2, 1], [8, 18, 2, 2], [8, 36, 2, 2], [8, 54, 2, 1], [8, 108, 2, 2], [12, 12, 1, 1], [12, 12, 2, 1], [12, 36, 1, 1], [12, 48, 1, 1], [12, 48, 2, 1], [12, 72, 1, 2], [12, 96, 1, 1], [12, 96, 2, 1], [12, 144, 1, 2], [12, 144, 2, 1], [24, 12, 2, 1], [24, 36, 2, 1], [24, 48, 2, 1], [24, 72, 2, 2], [24, 96, 2, 1], [24, 144, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^3\\times A_4^2.C_2^2\\times S_3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '3456.jo', 'autcentquo_hash': 2520327345672601111, 'autcentquo_nilpotent': False, 'autcentquo_order': 3456, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\times S_4^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 4], [2, 6, 2], [2, 9, 2], [2, 12, 1], [2, 18, 2], [2, 36, 2], [2, 72, 1], [2, 108, 1], [2, 216, 1], [3, 2, 1], [3, 8, 2], [3, 16, 2], [3, 64, 1], [3, 128, 1], [4, 6, 3], [4, 18, 4], [4, 36, 2], [4, 54, 1], [4, 72, 1], [4, 108, 2], [4, 216, 1], [6, 2, 1], [6, 6, 2], [6, 8, 2], [6, 12, 2], [6, 16, 2], [6, 24, 7], [6, 48, 4], [6, 64, 1], [6, 72, 3], [6, 96, 3], [6, 128, 1], [6, 144, 3], [6, 192, 3], [6, 288, 1], [8, 6, 2], [8, 18, 4], [8, 36, 4], [8, 54, 2], [8, 108, 4], [12, 12, 3], [12, 36, 1], [12, 48, 3], [12, 72, 2], [12, 96, 3], [12, 144, 4], [24, 12, 2], [24, 36, 2], [24, 48, 2], [24, 72, 4], [24, 96, 2], [24, 144, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '3456.jo', 'commutator_count': 1, 'commutator_label': '864.4195', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 209, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [['24.12', 1], ['48.29', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 4], [2, 6, 1, 2], [2, 9, 1, 2], [2, 12, 1, 1], [2, 18, 1, 2], [2, 36, 1, 2], [2, 72, 1, 1], [2, 108, 1, 1], [2, 216, 1, 1], [3, 2, 1, 1], [3, 8, 1, 2], [3, 16, 1, 2], [3, 64, 1, 1], [3, 128, 1, 1], [4, 6, 1, 3], [4, 18, 1, 4], [4, 36, 1, 2], [4, 54, 1, 1], [4, 72, 1, 1], [4, 108, 1, 2], [4, 216, 1, 1], [6, 2, 1, 1], [6, 6, 1, 2], [6, 8, 1, 2], [6, 12, 1, 2], [6, 16, 1, 2], [6, 24, 1, 7], [6, 48, 1, 4], [6, 64, 1, 1], [6, 72, 1, 3], [6, 96, 1, 3], [6, 128, 1, 1], [6, 144, 1, 3], [6, 192, 1, 3], [6, 288, 1, 1], [8, 6, 2, 1], [8, 18, 2, 2], [8, 36, 2, 2], [8, 54, 2, 1], [8, 108, 2, 2], [12, 12, 1, 3], [12, 36, 1, 1], [12, 48, 1, 3], [12, 72, 1, 2], [12, 96, 1, 3], [12, 144, 1, 4], [24, 12, 2, 1], [24, 36, 2, 1], [24, 48, 2, 1], [24, 72, 2, 2], [24, 96, 2, 1], [24, 144, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 2419200, 'exponent': 24, 'exponents_of_order': [8, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 0, 4], [24, 1, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '3456.jo', 'hash': 4930085545709402789, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 3, 4, 2, 3, 4, 3, 12, 6, 2], 'inner_gens': [[7620480, 40279680, 123381876, 87463545, 87178291200, 120113280, 6354061729, 174356582400, 174476333546, 93445628613, 127370880], [83462400, 127370880, 7260276, 80205945, 87178291200, 123379200, 6274228129, 174356582400, 174476333546, 93532719813, 40279680], [362880, 87091200, 43548276, 119773844, 87178291200, 3991680, 6270947148, 174356582400, 174483592212, 93492440133, 127370880], [39916800, 127370880, 123405310, 80205945, 87178291200, 43545600, 6274218644, 174356582400, 174483626639, 93532719813, 87091200], [7620480, 127370880, 43548276, 80205945, 87178291200, 47537280, 180630810529, 93405312000, 93449221226, 174483990213, 40279680], [123742080, 40279680, 7260276, 127017465, 87178291200, 47537280, 6314507809, 174356582400, 174403757546, 93445628613, 87091200], [362880, 127370880, 47544131, 80207225, 180583603200, 7257600, 6274228129, 93405312000, 93492780567, 174483990213, 87091200], [7620480, 127370880, 43548276, 80205945, 180583603200, 47537280, 87225498529, 174356582400, 174400491626, 93532719813, 40279680], [83462400, 40279680, 3994874, 87464825, 180583603200, 43545600, 87305328484, 174356582400, 174400491626, 93445628613, 127370880], [83462400, 127370880, 7260276, 80205945, 6227020800, 123379200, 180630810529, 174356582400, 174476333546, 93532719813, 40279680], [83462400, 127370880, 120115956, 47183865, 87178291200, 79833600, 6307250209, 174356582400, 174476333546, 93532719813, 40279680]], 'inner_hash': 2520327345672601111, 'inner_nilpotent': False, 'inner_order': 3456, 'inner_split': True, 'inner_tex': 'S_3\\times S_4^2', 'inner_used': [1, 2, 3, 4, 5, 7, 9], 'irrC_degree': 12, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 8], [2, 20], [3, 16], [4, 18], [6, 24], [8, 7], [9, 8], [12, 12], [16, 1], [18, 4], [24, 2]], 'label': '6912.ia', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': None, 'name': 'D6.S4^2', 'ngens': 11, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 87, 'number_characteristic_subgroups': 67, 'number_conjugacy_classes': 120, 'number_divisions': 105, 'number_normal_subgroups': 83, 'number_subgroup_autclasses': 3676, 'number_subgroup_classes': 4918, 'number_subgroups': 150202, 'old_label': None, 'order': 6912, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 559], [3, 242], [4, 720], [6, 2438], [8, 768], [12, 1224], [24, 960]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[7620480, 127370880, 43548276, 80205945, 87178328133, 47537280, 6274218644, 174356582400, 174400491626, 93532719813, 40279680], [7620480, 127370880, 43548276, 80205945, 87178291200, 47537280, 6274228129, 174356582400, 174400527119, 93532719813, 40279680], [7657413, 127370880, 43548276, 80207225, 87178291200, 47537280, 6274218644, 174356582400, 174400491626, 93532719813, 40279680]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': 7, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [3, 16], [4, 14], [6, 16], [8, 9], [9, 8], [12, 12], [16, 2], [18, 4], [24, 4]], 'representations': {'PC': {'code': '6187846104526929576604826054350833158790511891949585007462807337808917445735436437638288777550608838337972965665', 'gens': [1, 2, 4, 5, 7, 8, 9], 'pres': [11, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 3, 51173, 56, 157478, 6867, 3446, 1609, 70965, 63881, 158, 3184, 479, 3031, 88722, 19059, 10094, 1140, 199603, 13120, 5409, 1471, 294, 26453, 328, 23286]}, 'Perm': {'d': 15, 'gens': [7620480, 127370880, 43548276, 80205945, 87178291200, 47537280, 6274228129, 174356582400, 174400491626, 93532719813, 40279680]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'D_6.S_4^2', 'transitive_degree': 72, 'wreath_data': None, 'wreath_product': False}
